The impact of the streaming between baryons and dark matter on the first structures has been actively explored by recent studies. We investigate how the key results are affected by two popular approximations. One is to implement the streaming by accounting for only the relative motion while assuming ``baryons trace dark matter spatially at the initialization of simulation. This neglects the smoothing on the gas density taking place before the initialization. In our simulation initialized at $z_i=200$, it overestimates the gas density power spectrum by up to 40% at $kapprox10^2~h~mbox{Mpc}^{-1}$ at $z=20$. Halo mass ($M_h$) and baryonic fraction in halos ($f_{b,h}$) are also overestimated, but the relation between the two remains unchanged. The other approximation tested is to artificially amplify the density/velocity fluctuations in the cosmic mean density to simulate the first minihalos that form in overdense regions. This gives a head start to the halo growth while the subsequent growth is similar to that in the mean density. The growth in a true overdense region, on the other hand, is accelerated gradually in time. For example, raising $sigma_8$ by 50% effectively transforms $zrightarrowsqrt{1.5}z$ in the halo mass growth history while in 2-$sigma$ overdensity, the growth is accelerated by a constant in redshift: $zrightarrow{z+4.8}$. As a result, halos have grown more in the former than in the latter before $zapprox27$ and vice versa after. The $f_{b,h}$-$M_h$ relation is unchanged in those cases as well, suggesting that the Pop III formation rate for a given $M_h$ is insensitive to the tested approximations.
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